17.4 Ex 16-23

17.4 Ex 16-23 - S E C T I O N 17.4 Parametrized Surfaces...

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SECTION 17.4 Parametrized Surfaces and Surface Integrals (ET Section 16.4) 1117 = (( 3 + sin v) cos u ) i + (( 3 + sin sin u ) j (( 3 + sin cos k Hence, k n ( u ,v) k= ( 3 + sin p 1 + cos 2 v We obtain the following area: Area ( S ) = ZZ D k n k dud v = Z 2 π 0 Z 2 0 ( 3 + sin p 1 + cos 2 v v 144 . 0181 16. Let S be the surface z = ln ( 5 x 2 y 2 ) for 0 x , y 1. Using a computer algebra system: (a) Calculate the surface area of S to four decimal places. (b) Calculate S x 2 y 3 dS to four decimal places. SOLUTION (a) Using that z x =− 2 x /( 5 x 2 y 2 ) and z y 2 y /( 5 x 2 y 2 ) , we calculate k n k to be k n q 1 + ( z x ) 2 + ( z y ) 2 = 2 p x 2 + y 2 5 x 2 y 2 Thus, the surface area is Area ( S ) = Z 1 0 Z 1 0 2 p x 2 + y 2 5 x 2 y 2 dx dy 0 . 3698 (b) We calculate S x 2 y 3 as follows: S x 2 y 3 = Z 1 0 Z 1 0 x 2 y 3 2 p x 2 + y 2 5 x 2 y 2 0 . 0508 17. Use spherical coordinates to compute the surface area of a sphere of radius R . The sphere of radius R centered at the origin has the following parametrization in spherical coordinates: 8( θ , φ ) = ( R cos sin , R sin sin , R cos ), 0 2 , 0 The length of the normal vector is: k n R 2 sin Using the integral for surface area gives: Area ( S ) = D k n k d d = Z 2 0 Z 0 R 2 sin d d = à Z 2 0 R 2 d ± µ Z 0 sin d = 2 R 2 · µ cos ¯ ¯ ¯ ¯ 0 = 2 R 2 · 2 = 4 R 2 18. Compute the integral of z over the upper hemisphere of a sphere of radius R centered at the origin. The upper hemisphere has the following parametrization in spherical coordinates: , ) = ( R cos sin , R sin sin , R cos ), 0 2 , 0 2 The length of the normal vector is: k n R 2 sin The function z expressed in terms of the parameters is z = R cos , therefore we obtain the following surface integral, S zdS = Z 2 0 Z / 2 0 R cos · R 2 sin d d = Z 2 0 Z / 2 0 R 3 2 sin 2 d d = à Z 2 0 R 3 2 d ±Ã Z / 2 0 sin 2 d ± = R 3 2 · 2 µ cos 2 2 ¯ ¯ ¯ ¯ ¯ / 2 0 = R 3 µ 1 + 1 2 = R 3
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1118 CHAPTER 17 LINE AND SURFACE INTEGRALS (ET CHAPTER 16) 19. Compute the integral of x 2 over the octant of the unit sphere centered at the origin, where x , y , z 0.
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This homework help was uploaded on 04/13/2008 for the course MATH 32B taught by Professor Rogawski during the Winter '08 term at UCLA.

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17.4 Ex 16-23 - S E C T I O N 17.4 Parametrized Surfaces...

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